Universal operator growth and emergent hydrodynamics in quantum systems

I will present a hypothesis on universal properties of operators evolving under many-body Hamiltonian dynamics. I will define a measure of operator complexity and argue that it generically grows exponentially in time, with an exponent α, measurable through the properties of a physical retarded correlation function. Furthermore, the complexity exponent places a sharp bound on Lyapunov exponents λ ≤ 2α characterizing chaos, generalizing the known universal low-temperature bound λ ≤ 2πT. In a sense the complexity growth exponent offers a measure of chaos that does not rely on having a nearby semiclassical limit. I will illustrate the results in paradigmatic examples such as non-integrable quantum spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally I will present applications of this approach to computation of Hydrodynamic transport coefficients and for characterizing the many-body localization transition in disordered systems.